Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-9x+7y &= -4 \\ 5x-5y &= 6\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $5$ and the bottom equation by $7$ $\begin{align*}-45x+35y &= -20\\ 35x-35y &= 42\end{align*}$ Add the top and bottom equations. $-10x = 22$ Divide both sides by $-10$ and reduce as necessary. $x = -\dfrac{11}{5}$ Substitute $-\dfrac{11}{5}$ for $x$ in the top equation. $-9( -\dfrac{11}{5})+7y = -4$ $\dfrac{99}{5}+7y = -4$ $7y = -\dfrac{119}{5}$ $y = -\dfrac{17}{5}$ The solution is $\enspace x = -\dfrac{11}{5}, \enspace y = -\dfrac{17}{5}$.